How to make Eevee viewport background transparent?
You can use chosen color for EEVEE viewport using Light Path trick. 

In the Shader Editor, go to World view and mix HDRI Background with Solid Color Background using Is Camera Ray. This way your HDRI will have influence over the scene, but it will not be visible in viewport/camera.

To make it transparent you can set Alpha in Solid Background Color to 0 but it will be pure black color then.



If you want to render with actual transparent background you will need to set Film > Alpha to Transparent: